The given binary operation is a*b = a+b+2

In order to find the inverse of the relation, we have to find the identity element first.

Let that identity element be e then

a*e = a

From que.

a*e = a+e+2

So, from the above two relations we have

a+e+2 = a

or, e+2 = 0

∴ e = -2

Hence the identity element is -2 for this binary operation.

Now let a’ be the inverse of this relation

Then as per the definition of the inverse element

a*a’ = e

∴ a+a’+2 = -2

∴ a’ = -4 -a

And for 4 ,i.e. a = 4

a’ = -4 – 4

∴ a’ = -8

Thus the inverse element of 4 is -8 for the given binary operation.